28 July 2024
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Frequentist Bayesian
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Probability Long-run frequency Degree of belief
$P(D|H)$ $P(H|D)$
Parameters Fixed, true Random,
distribution
Obs. data One possible Fixed, true
Inferences Data Parameters
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n: 10
Slope: -0.1022
t: -2.3252
p: 0.0485
n: 10
Slope: -10.2318
t: -2.2115
p: 0.0579
n: 100
Slope: -10.4713
t: -6.6457
p: 1.7101362^{-9}
| Population A | Population B | |
|---|---|---|
| Percentage change | 0.46 | 45.46 |
| Prob. >5% decline | 0 | 0.86 |
\[ \begin{aligned} P(H\mid D) &= \frac{P(D\mid H) \times P(H)}{P(D)}\\[1em] \mathsf{posterior\\belief\\(probability)} &= \frac{likelihood \times \mathsf{prior~probability}}{\mathsf{normalizing~constant}} \end{aligned} \]
\[ \begin{aligned} P(H\mid D) &= \frac{P(D\mid H) \times P(H)}{P(D)}\\ \mathsf{posterior\\belief\\(probability)} &= \frac{likelihood \times \mathsf{prior~probability}}{\mathsf{normalizing~constant}} \end{aligned} \]
The normalizing constant is required for probability - turn a frequency distribution into a probability distribution
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\(P(D\mid H)\) |
subjectivity?
intractable \[P(H\mid D) = \frac{P(D\mid H) \times P(H)}{P(D)}\]
\(P(D)\) - probability of data from all possible hypotheses
Marchov Chain Monte Carlo sampling
Marchov Chain Monte Carlo sampling
Marchov Chain Monte Carlo sampling
Marchov Chain Monte Carlo sampling
Marchov Chain Monte Carlo sampling
Marchov Chain Monte Carlo sampling
Marchov Chain Monte Carlo sampling
Metropolis-Hastings
http://twiecki.github.io/blog/2014/01/02/visualizing-mcmc/
https://chi-feng.github.io/mcmc-demo/app.html?algorithm=GibbsSampling&target=banana
Gibbs
NUTS